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Any inﬁnite sequence of diﬀerent elements in an ordered set includes an inﬁnite monotone subsequence. 2. Any two open subintervals of R are similar. 3. Give an ordered set with a smallest element, in which every element has a successor and every element but the least has a predecessor, yet the set is not similar to N. 4. Give an ordering on the reals for which every element has a successor, as well as a predecessor. 5. An inﬁnite ordered set A, ≺ is similar to N if and only if for every a ∈ A there are only ﬁnitely many elements b ∈ A with b ≺ a.

Numbered by ordinals α. Here ω0 = ω is the smallest inﬁnite cardinal, and this numbering 52 Chapter 10 : Cardinals Problems is done so that β < α implies ωβ < ωα . , the smallest cardinal larger than κ), and is denoted by κ+ . It is always a regular cardinal. For historical reasons we also write ℵα instead of ωa (note that ωα has two faces; it is an ordinal and also a cardinal, and we use the aleph notation when we emphasize the cardinal aspect). , that there is no cardinal between ω and c) can be expressed as c = ℵ1 or as 2ℵ0 = ℵ1 .